Sheldon Lee's Research
Much of my past work involves numerical approximations of elliptic partial differential equations using variational analysis. Please email me at firstname.lastname@example.org for more information about past and current research.
Undergraduate Research at Viterbo: If you would like to take on an undergraduate research topic, please feel free to stop to get some possible ideas.
Possible undergraduate research topics:
- Developing efficient methods for representing the uncertainties in quantities computed from solutions of elliptic problems with randomly perturbed data and coefficients. These uncertainties often represent error in the mathematical model. The analysis could also apply to atomistic-to-continuum coupling, as the quantities computed from the atomistic region are uncertain. (See Atomistic to Continuum Coupling)
- Study the error passed back and forth in an atomistic to continuum coupling method. Rather than using only the expected value of the atomistic quantity as an input into the finite element model, control the stochastic error (standard deviation) by increasing the number of samples. In the continuum problem the basic errors can be divided into two categories: the uncertainty arising from the atomistic problem, and the discretization error arising from solving the problem using the finite element method. How do we account and correct for both types of errors in an appropriate balance? (See Atomistic to Continuum Coupling)
- Study error control for other types of coupling or in operator decomposition.
- Optimize functions with stochastic parameters. (See Adaptive Optimization)